By the time of the Star Wars films almost every species has access to space travel and faster-than-light technology. Has it ever been mentioned where this came from, did a specific species go out and teach other species it came across, did several species discover it at around the same time?
The purrgil were whale-like creatures that were naturally capable of entering hyperspace, and observation of these creatures inspired the development of the first hyperdrives. However, it's not clear exactly who developed the first hyperdrives or how the technology spread throughout the galaxy. It's likely the case that one or possibly a few civilizations invented the hyperdrive and then spread the technology to other civilizations as they used it to explore the galaxy.
The species most responsible for the spread of faster-than-light travel was probably the Rakata. Their hyperdrives were Force-powered and inspired by the Kwa species, but in creating their Infinite Empire the Rakata conquered many of the most important worlds of the galaxy (such as Coruscant, Corellia, the Wookiee homeworld of Kashyyyk, Tatooine, Dantooine, Manaan, etc.). When the Infinite Empire fell, some of the subject species were able to modify the Rakatan hyperdrive so that it no longer required the Force to operate. Space travel as we know it from the Star Wars films was possible at this point with star systems' connections from the fallen Infinite Empire and the means to travel faster-than-light and without the Force.
Since the Old Republic is said to have existed for a thousand generations, the invention of space travel/FTL should be so far in the past that probably no one knows or even is interested in, perhaps it was discovered independently on several worlds by different species.
Perhaps similar to the question when and where in Terran history the wheel was invented and how it spread its way across the world.